Hodge ideals of free divisors

TL;DR

This paper studies the Hodge filtration on meromorphic functions along free divisors, linking it to Bernstein-Sato polynomials, and develops an algorithm for computing Hodge ideals in this context.

Contribution

It describes the Hodge filtration steps via Bernstein-Sato polynomial factors and introduces an algorithm for computing Hodge ideals of free divisors.

Findings

Hodge filtration steps are characterized by Bernstein-Sato polynomial factors

An algorithm for computing Hodge ideals is developed and demonstrated

Conjecture on the bound for the generating level of the Hodge filtration

Abstract

We consider the Hodge filtration on the sheaf of meromorphic functions along free divisors for which the logarithmic comparison theorem holds. We describe the Hodge filtration steps as submodules of the order filtration on a cyclic presentation in terms of a special factor of the Bernstein-Sato polynomial of the divisor and we conjecture a bound for the generating level of the Hodge filtration. Finally, we develop an algorithm to compute Hodge ideals of such divisors and we apply it to some examples.